It's the process of scanning the negative in hi-res (2k or 4k) and then working on the image scene-by-scene (and ususally shot-by-shot) with sophisticated image manipulation software which allows much more in-depth work than traditional color correction, working into specific hues and/or areas of the image. The end result is then usually printed back out to film (although George Lucas, among others, wants it all to be digitally projected one day). It is the computer-age replacement for optical color-timing, although it is possible to do a lot more with a DI, with the consequence that everyone usually does a lot more (see Lord of the Rings et al).